Extension beyond affine public communication and linear candidates

Extend the recovery comparison for six-state quantum key distribution beyond affine public communication and linear candidate keys, thereby determining whether the same impossibility result holds for nonlinear public processing or directly shared nonlinear outputs.

Background

The main theorem applies to protocols whose public messages are affine functions of the original raw keys and whose Alice-side candidate key is a linear or affine function selected from the public transcript. Nonlinear Bob decoding is allowed, and nonlinear privacy amplification is covered only after reliable recovery of a linear candidate.

The authors explicitly leave open whether the recovery comparison and resulting limitation at p_* can be extended to protocols using broader nonlinear public communication or directly shared nonlinear outputs. This is a genuine boundary of the proved converse rather than an issue addressed by the theorem.